Armature doesn't animate despite being keyframe'd (video)
Constraints are applied after keyframed values. In a similar fashion the values that normally get keyframed are before constraints have been applied.

First you will need to disable the constraint to see the keyframed animation as the constraint will hide any movement if it's source is not animated or it's animation is muted.

To get the result you want, you need to use one of the "Visual" keying sets, these will keyframe transformation values after constraints have been applied. With a constraint then disabled you should see the animation that you are aiming for.